Question

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The angles of a triangle measure 2x , 3x , and 4x . Using the Triangle Angle Sum Theorem, find the measure of the smallest angle in this triangle.(1 point)
Answer
Use the image to answer the question.

An illustration shows a triangle with vertices as B C D. Angle C on the top is labeled as a question mark and angle D on the bottom right is labeled as 22 degrees. Vertex B is on the bottom left. The horizontal line D B extends beyond the triangle to the left with an arrow at the end, and a point A is marked on it. The angle of A B C is labeled as 103 degrees.

Use the Triangle Angle Sum Theorem to find the measure of the angle on point C .

(1 point)
Answer
Use the Triangle Angle Sum Theorem to find the largest angle in a triangle with angle measures of (x−20)° , (3x+3)° , and (2x−7)° .(1 point)
Answer
Use the image to answer the question.

A triangle is shown with its angles measuring 2 x, 96 degrees, and x plus 12 degrees.

Find the measure of the smallest angle of the triangle.(1 point)
°
Answer
Two sides of a triangle are 10 mm and 7 mm. Determine the length of the third side from the choices given.(1 point)
Responses

20 mm
20 mm

1 mm
1 mm

5 mm
5 mm

2 mm
Answer
Complete the condition statements that must be met in order for three side lengths— a , b , and c —to create a triangle.(1 point)
a
b+c and a
b−c
Answered by GPT 3.5
The sum of the three angles of a triangle is always 180 degrees.

Therefore, we can set up the equation:

2x + 3x + 4x = 180

Simplifying the equation, we get:

9x = 180
x = 20

Now we can find the measure of the smallest angle by substituting x back into the smallest angle measurement:

2(20) = 40 degrees

Therefore, the measure of the smallest angle in this triangle is 40 degrees.
Answered by GPT 3.5
The Triangle Angle Sum Theorem states that the sum of the angles in a triangle is always 180 degrees.

We are given that angle D is 22 degrees and angle B is 103 degrees. To find the measure of angle C, we can add these two angles together and subtract the sum from 180 degrees:

Angle C = 180 - (angle D + angle B)
Angle C = 180 - (22 + 103)
Angle C = 180 - 125
Angle C = 55 degrees

Therefore, the measure of angle C is 55 degrees.
Answered by GPT 3.5
The Triangle Angle Sum Theorem states that the sum of the angles in a triangle is always 180 degrees.

Given angles in the triangle:
1. (x - 20)°
2. (3x + 3)°
3. (2x - 7)°

Adding these angles and setting the sum equal to 180 degrees, we get:

(x - 20) + (3x + 3) + (2x - 7) = 180
Combine like terms:
6x - 24 = 180
6x = 204
x = 34

Now we can find the measure of each angle by substituting the value of x back into each angle measure:
1. (x - 20) = 34 - 20 = 14°
2. (3x + 3) = 3(34) + 3 = 105°
3. (2x - 7) = 2(34) - 7 = 61°

The largest angle in the triangle is 105°.
Answered by GPT 3.5
To find the measure of the smallest angle in the triangle, we need to use the Triangle Angle Sum Theorem, which states that the sum of the angles in a triangle is always 180 degrees.

Given angles in the triangle:
1. 2x
2. 96 degrees
3. x + 12

According to the Triangle Angle Sum Theorem, the sum of these angles is 180 degrees:
2x + 96 + x + 12 = 180
3x + 108 = 180
3x = 72
x = 24

Now, substitute x back into each angle to find their measures:
1. 2x = 2(24) = 48 degrees
2. 96 degrees
3. x + 12 = 24 + 12 = 36 degrees

Therefore, the smallest angle of the triangle is 36 degrees.
Answered by GPT 3.5
To determine the length of the third side of the triangle, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given sides of the triangle:
1. 10 mm
2. 7 mm

Let x be the length of the third side. According to the triangle inequality theorem:
10 + 7 > x
17 > x

Therefore, the length of the third side must be less than 17 mm. Hence, the correct choice from the given options is:
5 mm

Because 10 + 7 > 5 is true, while 10 + 7 is not greater than 20 or 1.
Answered by GPT 3.5
In order for three side lengths a, b, and c to create a triangle, the following conditions must be met:

1. The sum of the lengths of any two sides must be greater than the length of the third side:
a + b > c
b + c > a
a + c > b

2. The difference of the lengths of any two sides must be less than the length of the third side:
|a - b| < c
|b - c| < a
|a - c| < b

These conditions ensure that three side lengths can form a valid triangle.